Finding the widest acceptable ranges for x of given y while meeting certain criteria

Question

I am thinking about a problem for some time now and it would be awesome if someone could give me advice how to solve it (I prefer python).

Given are 8 equitations (y1 to y8) with 5 independent variables x1, x2, x3, x4, x5. The functions look like this:

y1 = f(x1,x2,x3,x4,x5) = a*x1 + b*x2 + c*x3 + d*x4 + e*x5 + a*x1*x2 + b*b*x2 + c*c*x3
y2 = f(x1,x2,x3,x4,x5) = f*x1 + g*x2 + h*x5 + a*a*x1 + d*d*x4 ...
y3 = f(x1,x2,x3,x4,x5) = ...
...
y8 = f(x1,x2,x3,x4,x5) = i*x1 + k*x3 + l*x5 + j*j*x2*x2 + c*c*x3 + ...

As you can see they only share the same x1,x2,x3,x4,x5, but the coefficients are all different. The number of terms can be different too.

The range of each variable x allowed to be used in y is restricted to:

x1 = (1,10) #any float between 1 and 10 can be used
x2 = (10,20)
x3 = (100,200)
x4 = (50,80)
x5 = (20,50)

The goal is to find the 'widest' ranges for each of the 5 independent variables that will meet the following criteria: y1 must not by lower than 10, y2 must not by lower than 7, y3 must not by lower than 20 ... So basically each y has to yield a specified minium value. This means for every combination of x1,x2,x3,x4,x5 all 8 equitations need to be calculated and checked whether all y1 to y8 fulfill the criteria.

One solution for the 5 ranges could be:

x1 = (3,7)
x2 = (14,15)
x3 = (111,130)
x4 = (60,65)
x5 = (20,22)

but the following solution would be better because the ranges are wider:

x1 = (2,10) # 10-2 = 8. This is a wider range than the x1 example above (7-3 = 4)
x2 = (11,19)
x3 = (104,190)
x4 = (51,78)
x5 = (21,49)

How can I find the widest ranges for each x while meeting the y criteria? How would you solve this or at least try to find the best solution? Brute forcing all the combinations cannot be done in acceptable time.